Abstract

Given a family X of complex varieties degenerating over a punctured disk, one is interested in computing related invariants called the motivic nearby fiber and the refined limit mixed Hodge numbers, both of which contain information about the induced action of monodromy on the cohomology of a fiber of X. Our first main result is that the motivic nearby fiber of X can be computed by first stratifying X into locally closed subvarieties that are nondegenerate in the sense of Tevelev, and then applying an explicit formula on each piece of the stratification that involves tropical geometry. Our second main result is an explicit combinatorial formula for the refined limit mixed Hodge numbers in the case when X is a family of nondegenerate hypersurfaces. As an application, given a complex polynomial, then, under appropriate conditions, we give a combinatorial formula for the Jordan block structure of the action of monodromy on the cohomology of the Milnor fiber, generalizing a famous formula of Varchenko for the associated eigenvalues. In addition, we give a formula for the Jordan block structure of the action of monodromy at infinity.

Keywords

1 Introduction

Before introducing the main results of this paper, we first present an application. Let \(f(x_1, \ldots , x_n) \in \mathbb {C}[x_1, \ldots , x_n]\) be a complex polynomial with no constant term, and consider the induced map \(f: \mathbb {C}^n \rightarrow \mathbb {C}\). One would like to understand the singularity of \(f^{-1}(0) \subseteq \mathbb {C}^n\) at the origin (we assume this is an isolated singularity). A classical invariant used to distinguish the singularities arising from different polynomials is the action of monodromy on the cohomology of the associated Milnor fiber. In simple terms, this means associating to f a finite, square complex matrix \(M_f\) that is well defined up to change of basis. We assume that f is ‘general’ in the sense that it is nondegenerate with respect to its Newton polyhedron\(\Gamma _+(f)\), and also that it is convenient (see Sect. 6.3). The most famous result was proved by Varchenko in 1976 [61], who gave an explicit combinatorial formula for the eigenvalues (with multiplicity) of \(M_f\), involving alternating signs of volumes of polytopes associated with \(\Gamma _+(f)\). Unfortunately, since \(M_f\) is rarely diagonalizable, it is not, in general, determined by its eigenvalues. We are left with the problem of giving combinatorial formulas for the numbers \(J_{k,\alpha }\) of Jordan blocks of \(M_f\) of size k with eigenvalue \(\alpha \). Using tropical geometry, together with weighted Ehrhart theory, which involves the enumerative combinatorics of lattice points in dilates of polytopes, we settle this question:

Theorem 1.1

(Corollary 6.22) Let \(f \in \mathbb {C}[x_1,\ldots , x_n]\) be a complex polynomial such that \(f^{-1}(0)\) admits an isolated singularity at the origin. Assume further that f is convenient and nondegenerate with respect to its Newton polyhedron \(\Gamma _+(f)\). Then there are explicit, nonnegative combinatorial formulas for the numbers \(J_{k,\alpha }\), involving the weighted Ehrhart theory of polytopes associated with \(\Gamma _+(f)\).

We refer the reader to Corollary 6.22 for the specific formulas. By a ‘nonnegative’ formula, we mean that each \(J_{k,\alpha }\) is expressed as a sum of nonnegative integers. By summing over k, we obtain a nonnegative formula for the multiplicities of the eigenvalues of \(M_f\), which is equivalent to Varchenko’s formula (Remark 6.19). Using different techniques, special cases of formulas for the \(J_{k,\alpha }\) have previously been proved by Matsui and Takeuchi [40] and van Doorn and Steenbrink [60], and an algorithm to compute the numbers \(J_{k,\alpha }\) involving simplicial resolutions of toric varieties was proved by Matsui and Takeuchi in [40], extending work of Danilov [13] and Tanabé [58]. We note that algorithms in a more general setting are given by Schulze in [49].

The theorem above is a corollary of more general results that we will outline in the remainder of the introduction. Given a family of varieties \(f:X\rightarrow \mathbb {D}^*\) over a complex disk, Denef and Loeser [17] associate an invariant \(\psi _X\) called the motivic nearby fiber, which contains information about the variation of Hodge structures associated with the degeneration. In particular, if we fix a nonzero fiber \(X_{{{\mathrm{gen}}}} := f^{-1}(t)\) for some sufficiently small \(t \in \mathbb {D}^*\), then \(\psi _X\) encodes information on the induced monodromy map \(T: H^m_c(X_{{{\mathrm{gen}}}}) \rightarrow H^m_c(X_{{{\mathrm{gen}}}})\), which is a linear operator on the complex cohomology with compact supports of \(X_{{{\mathrm{gen}}}}\) and is quasi-unipotent, i.e., \(T = T_sT_u\), where \(T_s\) and \(T_u\) commute, \(T_s\) is semi-simple and has finite order, and \(T_u\) is unipotent. In [30], Eric Katz and the author showed that one can use tropical geometry to concretely compute a specialization of \(\psi _X\) that is invariant under base change, and hence encodes information about \(T_u\) but not \(T_s\). Using new combinatorics developed in [29], in the case of families of schön complex hypersurfaces of tori, explicit combinatorial formulas were then deduced for the refined limit mixed Hodge numbers, which, in particular, determine the Jordan block structure of the action of \(T_u\) on the graded pieces (with respect to the Deligne weight filtration) of \(H^m_c(X_{{{\mathrm{gen}}}})\). Here schön is a ‘generic’ condition introduced by Tevelev [59], generalizing the notion of nondegeneracy of a hypersurface of a complex torus [33]. Equivalent formulas were also given for the action of \(T_u\) on the intersection cohomology groups of a family of schön hypersurfaces of a projective toric variety.

The goal of this note is to extend the above results to the full generality of the motivic nearby fiber \(\psi _X\) and monodromy operator T. Our first main result is to show that the motivic nearby fiber may be computed using tropical geometry (see Sect. 3). A key point is that since there is a range of available software implementing the main algorithms in tropical geometry [26, 27], our main formula Theorem 3.2 can be computed in practice.

Weighted Ehrhart theory was introduced by the author in [52] both in order to extend and reprove many results in Ehrhart theory, the study of enumeration of lattice points in polytopes, and in order to give explicit computations of motivic integrals on toric stacks [53]. Roughly speaking, one associates with every lattice point v a ‘weight’ w(v) in \(\mathbb {Q}/\mathbb {Z}\) and then attempts to enumerate lattice points in polytopes keeping track of the associated weights. Extending work in [29], in Sect. 4.4, we associate new Ehrhart-theoretic invariants to a pair \((P,\nu )\), where P is a lattice polytope and \(\nu \) is the convex graph of an integral height function on P. In particular, we introduce the weighted refined limit mixed\(h^*\)-polynomial\(h^*(P,\nu ;u,v,w) \in \mathbb {Z}[\mathbb {Q}/\mathbb {Z}][u,v,w]\). Roughly speaking, our second main result states that if \(X^\circ \) is a family of schön complex hypersurfaces of tori, then the associated equivariant refined limit mixed Hodge numbers are precisely the coefficients of \(h^*(P,\nu ;u,v,w)\) (Theorem 5.7, Corollary 5.12), where \((P,\nu )\) is the Newton polytope and associated convex graph associated with \(X^\circ \). In particular, this gives combinatorial formulas for the Jordan block structure of the action of T on the graded pieces (with respect to the Deligne weight filtration) of \(H^m_c(X^\circ _{{{\mathrm{gen}}}})\). Equivalent formulas are also given for the equivariant refined limit mixed Hodge numbers associated with the intersection cohomology groups of a family of schön hypersurfaces of a projective toric variety (Theorem 5.9, Corollary 5.12). In Example 5.13, we express these invariants in a special case in terms of dimensions of orbifold cohomology [11, 12] of a toric stack [4]. It would be interesting to have an explanation of this fact involving ‘mirror symmetry’.

As a special case of the above result, in Theorem 5.11, we give a formula for the equivariant Hodge–Deligne polynomial of a schön complex hypersurface of a torus invariant under the action of an element of the torus of finite order. An algorithm to determine this polynomial had previously been given by Matsui and Takeuchi in [39], generalizing an algorithm of Danilov and Khovanskiĭ in the nonequivariant setting [14]. Moreover, the above formula reduces to a formula of Borisov and Mavlyutov [5] in the nonequivariant setting, which itself is a simplification of a formula of Batyrev and Borisov [3].

We next present some applications to the monodromy of complex polynomials. Let \(f(x_1, \ldots , x_n) \in \mathbb {C}[x_1, \ldots , x_n]\) be a complex polynomial. It is well known that there exists a minimal finite subset \(B_f \subseteq \mathbb {C}\), such that \(f: \mathbb {C}^n \rightarrow \mathbb {C}\) is a locally trivial fibration away from \(B_f\). Then monodromy defines an action of the fundamental group \(\pi _1(\mathbb {C}\smallsetminus B_f)\) on the cohomology of a fixed generic fiber of f. On the one hand, we are interested in the monodromy action induced by moving anti-clockwise around a small loop around an element of \(b \in B_f\). After translating f by a constant, we may assume that \(b = 0\), and then the resulting action is called monodromy at 0. On the other hand, choose \(R > 0\) sufficiently large such that \(B_f\) is strictly contained in \(\{ z \in \mathbb {C}\mid |z| = R \}\). Then the monodromy action on cohomology induced by moving clockwise around such a loop is called the monodromy at infinity of f, and a fundamental result of Dimca and Némethi [18] states that monodromy at infinity essentially determines the monodromy action of \(\pi _1(\mathbb {C}\smallsetminus B_f)\). We assume that f is convenient in the sense that its Newton polytope has nonzero intersection with each ray through a coordinate vector [34]. Then under certain schönness (also called ‘nondegeneracy’) conditions, we give explicit combinatorial formulas that essentially completely describe both monodromy at 0 (Example 6.7) and monodromy at infinity (Example 6.8). In particular, we deduce ‘nonnegative’ combinatorial formulas for both the equivariant limit mixed Hodge numbers associated with monodromy at infinity (Corollary 6.13) and the Jordan block structure of monodromy at infinity (Corollary 6.14). Algorithms to compute the latter invariants were given by Matsui and Takeuchi in [39]. Also, these formulas specialize to known formulas for the spectrum at infinity of f [39, Theorem 5.11] and zeta function at infinity of f [35]. We note that some of the above results may be extended without the assumption that f is convenient (see Remark 6.1). We also present analogous ‘local’ results for the Milnor fiber of a polynomial f with an isolated singularity at the origin, including combinatorial formulas for both the corresponding equivariant limit mixed Hodge numbers, and, as outlined at the beginning of this introduction, the Jordan block structure of the action of monodromy on the cohomology of the Milnor fiber.

Finally, we mention some possible generalizations of the above results that we expect will follow by similar methods. Firstly, using the ‘Cayley trick’ of Danilov and Khovanskiĭ [14, Section 6], one may extend our results on hypersurfaces to the case of complete intersections. In particular, in the case of the monodromy at infinity and monodromy of the cohomology of the Milnor fiber, one may obtain explicit formulas that extend algorithms given by Esterov and Takeuchi in [23]. Secondly, in the case when f is not convenient, one may use the results above to obtain formulas extending algorithms of Takeuchi and Tibar in [57]. Lastly, analogous formulas to those in Sect. 6.3 for families of hypersurfaces of an affine toric variety (rather than simply affine space) may be obtained using the setup and results of Steenbrink in [56].

1.1 Organization of the paper

This paper is structured as follows. In Sect. 2, we review the necessary geometric background. In Sect. 3, we show how to compute the motivic nearby fiber using tropical geometry. In Sect. 4, we review the necessary combinatorial background before introducing our main combinatorial invariants. In Sect. 5, we prove our combinatorial formulas for geometric invariants associated with degenerations of hypersurfaces. Finally, in Sect. 6, we apply our results to deduce formulas for invariants associated with the monodromy of complex polynomials.

2 Geometry of degenerations

In this section we briefly recall the notion of the motivic nearby fiber and its associated geometry. We refer the reader to Bittner [7], Peters [43] and Peters and Steenbrink [44] for details. A similar exposition is given in the nonequivariant setting in [30, Section 3].

2.1 The equivariant Grothendieck ring

We follow the treatment in [7] below. Fix a field k. The Grothendieck ring\(K_0({{\mathrm{Var}}}_k)\) of algebraic varieties over k is the free abelian group generated by isomorphism classes [V] of varieties V over k, modulo the relation

$$\begin{aligned}{}[V] = [U] + [V \smallsetminus U], \end{aligned}$$

(1)

whenever U is an open subvariety of V. Multiplication is given by tensor product of varieties.

Suppose that G is a finite group. An (algebraic) action of G on a variety over k is good if every orbit lies in an open affine subvariety. Then we may similarly define \(\widetilde{K}_0^{G}({{\mathrm{Var}}}_k)\) to be the free abelian group generated by isomorphism classes [V] of varieties V over k together with a good action of G, modulo the relation \([V] = [U] + [V \smallsetminus U]\), whenever U is a G-invariant open subvariety of V. The equivariant Grothendieck ring\(K_0^{G}({{\mathrm{Var}}}_k)\) is the quotient of \(\widetilde{K}_0^{G}({{\mathrm{Var}}}_k)\) by the relation \([\mathbb {P}(V) \circlearrowleft G] = [\mathbb {P}^n \times (X \circlearrowleft G)]\), where \(V \rightarrow X\) is a vector bundle of rank \(n + 1\) with a good G-action that is linear over a good G-action on X. Here \(\mathbb {P}(V)\) denotes the projectivization of V with induced G-action, and \(\mathbb {P}^n\) admits a trivial G-action. A group homomorphism \(G \rightarrow H\) induces a ring homomorphism \(K_0^{H}({{\mathrm{Var}}}_k) \rightarrow K_0^{G}({{\mathrm{Var}}}_k)\) by restriction. In particular, \(K_0^{G}({{\mathrm{Var}}}_k)\) is a module over \(K_0({{\mathrm{Var}}}_k)\), and forgetting the action of G gives a ring homomorphism \(K_0^{G}({{\mathrm{Var}}}_k) \rightarrow K_0({{\mathrm{Var}}}_k)\). We will follow the convention that \(\mathbb {L}:= [\mathbb {A}^1]\) is the class of the affine line with trivial G-action. A motivic invariant is a ring homomorphism \(K_0^{G}({{\mathrm{Var}}}_k) \rightarrow R\), for some ring R.

2.3 Motivic invariants

The importance of the motivic nearby fiber comes from its specializations. In this section, we will explain the following commutative diagram, which will be crucial in the rest of the paper, where the first vertical map is the motivic nearby fiber:

Remark 2.3

Consider the \(\mathbb {Z}\)-algebra involution on \(\mathbb {Z}[\mathbb {Q}/\mathbb {Z}]\) defined by setting \(\overline{[k]} = [-k]\), corresponding to complex conjugation on the circle. This extends coefficient-wise to an involution on a polynomial ring with coefficients in \(\mathbb {Z}[\mathbb {Q}/\mathbb {Z}]\) that we will refer to as conjugation.

Remark 2.4

Consider the natural \(\mathbb {Z}\)-algebra homomorphism \(\mathbb {Z}[\mathbb {Q}/\mathbb {Z}] \rightarrow \mathbb {Z}, [k] \mapsto 1\). One may apply this homomorphism below to get invariants with coefficients in \(\mathbb {Z}\) rather than \(\mathbb {Z}[\mathbb {Q}/\mathbb {Z}]\). The corresponding diagram of invariants above over \(\mathbb {Z}\) is explained in detail in [30, Section 3].

In [15], Deligne proved that the \(m^{\text {th}}\) cohomology group with compact supports \(H^m_c (V)\) of a complex variety V admits a canonical mixed Hodge structure with decreasing filtration \(F^\bullet \) called the Hodge filtration and increasing filtration \(W_\bullet \) called the Deligne weight filtration. Given a good action of \(\widehat{\mu }\) on V, we may set \(B_m = H^{p,q}(H^m_c (V))\) above. The corresponding equivariant Hodge polynomial is denoted \(E_{\widehat{\mu }}(V;u,v) \in \mathbb {Z}[\mathbb {Q}/\mathbb {Z}][u,v]\), and is called the equivariant Hodge–Deligne polynomial of V. The invariants \(h^{p,q}(H^m_c (V))_\alpha \) are called the equivariant mixed Hodge numbers of V. We have a ring homomorphism called the equivariant Hodge–Deligne map:

where R(G) is the complex representation ring of G. This induces a ring homomorphism \(\widetilde{K}_0^{G}({{\mathrm{Var}}}_\mathbb {C}) \rightarrow R(G)[u,v]\). This invariant was introduced and studied by the author in [54], generalizing the notion of weight polynomial due to Dimca and Lehrer [19], and the notion of equivariant \(\chi _y\)-genus due to Cappell, Maxim and Shaneson [10]. In our case, \(\mathbb {Z}[\mathbb {Q}/\mathbb {Z}]\) may be viewed as the direct limit associated with the restriction maps \(R(\mu _{n}) \rightarrow R(\mu _{nd})\) induced by the epimorphisms \(\mu _{nd} \rightarrow \mu _n, \zeta \mapsto \zeta ^d\).

Recall that a variety X over \(\mathbb {K}\) is naturally interpreted as a complex variety with a morphism \(f:X\rightarrow \mathbb {C}^* \smallsetminus \{ b_1, \ldots , b_r \}\), for some points \(b_1,\ldots , b_r\). Restricting X to a family over a sufficiently small punctured complex disk centered at the origin, and fixing a fiber \(X_{{{\mathrm{gen}}}} := f^{-1}(t)\) and counter-clockwise orientation around the disk, there exists a quasi-unipotent monodromy map \(T = T_sT_u: H^m_c(X_{{{\mathrm{gen}}}}; \mathbb {C}) \rightarrow H^m_c(X_{{{\mathrm{gen}}}}; \mathbb {C})\), where \(T_s\) is semi-simple and \(T_u\) is unipotent. Then the cohomology groups \(H_c^m(X_{{{\mathrm{gen}}}})\) admit a weight filtration \(M_\bullet \) called the monodromy weight filtration, and \(T_s\) acts preserving the filtrations \((F^\bullet ,M_\bullet , W_\bullet )\). We will write \(H_c^m(X_{\infty })\) to denote \(H_c^m(X_{{{\mathrm{gen}}}})\) with the filtrations \((F^\bullet ,M_\bullet ,W_\bullet )\). The filtrations \((F^\bullet ,M_\bullet )\) induce a mixed Hodge structure on \(H_c^m (X_{\infty })\), and the nilpotent operator \(N = \log T_u\) is a morphism of mixed Hodge structures of type \((-1,-1)\). The corresponding invariants \(h^{p,q}(H_c^m (X_{\infty }))_\alpha \) are called the equivariant limit mixed Hodge numbers, and the corresponding equivariant Hodge polynomial \(E(X_\infty , \widehat{\mu };u,v) \in \mathbb {Z}[\mathbb {Q}/\mathbb {Z}][u,v]\) is called equivariant limit Hodge–Deligne polynomial. A deep result of Denef and Loeser states that this is a specialization of the motivic nearby fiber in the sense that \(E(X_\infty ,\widehat{\mu };u,v) = E_{\widehat{\mu }}(\psi _X)\). As in Remark 2.4, we may consider the corresponding invariants over \(\mathbb {Z}\), i.e., the limit mixed Hodge numbers and limit Hodge–Deligne polynomial.

Before proceeding, we recall the following standard linear algebra construction.

Definition 2.6

Let A be a nilpotent linear operator on a finite-dimensional vector space V such that \(A^{r + 1} = 0\). Then the A-weight filtration centered atr is the increasing filtration \(\{ V_\bullet \}\) of V by subspaces

The monodromy weight filtration \(M_\bullet \) has the following important geometric property: for every nonnegative integer r, the induced filtration \(M(r)_\bullet \) on \(Gr_r^W H_c^m (X_{\infty })\) coincides with the N(r)-weight filtration centered at r, where N(r) is the nilpotent operator induced by \(N = \log T_u\). It follows from Definition 2.6 that the monodromy weight filtration, together with the action of \(T_s\), encodes the Jordan block structure of the monodromy operator T acting on \(Gr_r^W H_c^m (X_{\infty })\). Moreover, \((F^\bullet ,M_\bullet )\) induces a mixed Hodge structure on \(Gr_r^W H_c^m (X_{\infty })\) that is invariant under \(T_s\), and N(r) is a morphism of mixed Hodge structures of type \((-1,-1)\). The associated equivariant mixed Hodge numbers

are called the equivariant refined limit mixed Hodge numbers, and the associated equivariant Hodge polynomial is the coefficient of \(w^r\) in a polynomial \(E(X_\infty , \widehat{\mu };u,v,w) \in \mathbb {Z}[\mathbb {Q}/\mathbb {Z}][u,v,w]\) called the equivariant refined limit Hodge–Deligne polynomial. As in Remark 2.4, we have corresponding invariants over \(\mathbb {Z}\) that were introduced in [30, Section 3], to where we refer the reader for further details. We have a ring homomorphism

Hence the equivariant refined limit Hodge–Deligne polynomial satisfies the following symmetries with respect to the conjugation action on \(\mathbb {Z}[\mathbb {Q}/\mathbb {Z}][u,v,w]\) described in Remark 2.3:

Example 2.9

If V is a complex variety and \(X = V \times _\mathbb {C}\mathbb {K}\), then X may be regarded as a trivial family over \(\mathbb {D}^*\). In this case, N is identically zero, \(M_\bullet \) coincides with the Deligne filtration \(W_\bullet \), and \(E(X_\infty ,\widehat{\mu };u,v,w) = E(V;uw,vw)\).

Example 2.10

If X is smooth and proper, then \(Gr_r^W H^m (X_{{{\mathrm{gen}}}}) = 0\) unless \(m = r\). In this case, the monodromy weight filtration, together with the action of \(T_s\), encodes the Jordan block decomposition of T. In this case,

3 Tropical geometry

In this section we prove our formula for the motivic nearby fiber. The nonequivariant version of this result is [30, Theorem 1.2]. We continue with the notation of the previous section. In particular, \(\mathbb {K}= \mathbb {C}(t)\).

A variety X over \(\mathbb {K}\) is very affine if it admits a closed embedding \(X \subseteq (\mathbb {K}^*)^n\) for some n. Tevelev introduced the notion of a schön, very affine variety. A closed subvariety \(X^\circ \subseteq (\mathbb {K}^*)^n\) is schön if \({{\mathrm{in}}}_w X^\circ \subseteq (\mathbb {C}^*)^n\) is a smooth (reduced) subvariety for every \(w \in \mathbb {R}^n\) [25, Prop 3.9]. A very affine variety X over \(\mathbb {K}\) is schön if it admits a schön closed embedding \(X \subseteq (\mathbb {K}^*)^n\) for some n (cf. [37, Lemma 2.11]). Note that a complex variety V may be viewed via base change as a variety over \(\mathbb {K}\), and hence we may consider the same notions. The definition of schön for a hypersurface of a complex torus agrees with the notion of nondegeneracy with respect to the Newton polytope [33].

The tropical variety\({{\mathrm{Trop}}}(X^\circ )\) associated with \(X^\circ \subseteq (\mathbb {K}^*)^n\) is the set of points

The tropical variety \({{\mathrm{Trop}}}(X^\circ )\) can be given a rational polyhedral structure \(\Sigma \) such that initial degeneration at \(w \in {{\mathrm{Trop}}}(X^\circ )\) only depends on the cell containing w in its relative interior (this follows from [37, Theorem 1.5]). Hence for every cell \(\gamma \) of \(\Sigma \), we may define \({{\mathrm{in}}}_{\gamma } X^\circ := {{\mathrm{in}}}_w X^\circ \) for any \(w \in \mathbb {R}^n\) in the relative interior of \(\gamma \). Moreover, by fixing such a \(w \in \mathbb {Q}^n\), we obtain a good action of \(\widehat{\mu }\) on \({{\mathrm{in}}}_{\gamma } X^\circ \) via multiplication by \(\exp ( 2\pi \sqrt{-1} w)\).

A result of Luxton and Qu [37, Theorem 6.11] that was conjectured by Tevelev in [59], implies that every variety over \(\mathbb {K}\) admits a stratification into locally closed, very affine, schön subvarieties. By additivity property (1) of \(K_0({{\mathrm{Var}}}_\mathbb {K})\), it follows that we may reduce the computation of \(\psi \) to the case when \(X^\circ \subseteq (\mathbb {K}^*)^n\) is a schön subvariety. In this case, we have the following equivariant generalization of [30, Theorem 1.2].

where the vertical arrows are flat. After removing the closed torus invariant divisors corresponding to nonzero cones in \(\Delta \), the nonzero fibers of \(\widetilde{\mathbb {P}}\) and \(\mathbb {P}\) are tori and the central fibers have decompositions into disjoint unions of tori:

Recall that \(X^\circ \) may viewed as a family of complex varieties over \(\mathbb {C}^* \smallsetminus \{ b_1, \ldots , b_r \}\), for some points \(b_1,\ldots , b_r\). Let \(\mathbb {D}\) be a sufficiently small complex disk centered at the origin, and restrict \(X^\circ \) to a family over the punctured disk \(\mathbb {D}^*\), and \(\widetilde{\mathbb {P}}\) and \(\mathbb {P}\) to families over \(\mathbb {D}\). Taking the closure \({\mathcal {X}}\) of \(X^\circ \) in \(\mathbb {P}\) induces a commutative diagram:

where \(\widehat{\mu }\) acts via multiplication by \(\exp (2\pi \sqrt{-1} w)\) and \(\exp (2\pi \sqrt{-1} [-\gamma ])\), respectively. In particular, this shows that the left-hand side is independent of the choice of w in the relative interior of \(\gamma \).

We now complete the proof. We first verify that the right-hand side of the statement in the theorem is independent of the choice of polyhedral structure \(\Sigma \). Indeed, if \(\Sigma '\) is a polyhedral structure refining \(\Sigma \), then, using the independence of the choice of w above

Remark 3.3

It is a corollary of the proof above that in the statement of Theorem 3.2, one may consider the action on \({{\mathrm{in}}}_\gamma X^\circ \subseteq (\mathbb {C}^*)^n\) to be induced by multiplication by \(\exp (2\pi \sqrt{-1} w)\) for any fixed choice of \(w \in \mathbb {Q}^n\) in the affine span of \(\gamma \).

We will see interesting examples of the above theorem in Sect. 5 and Sect. 6. For the moment, we have the following simple example.

4 The combinatorics of height functions

In this section, we present the relevant combinatorics of this paper. Weighted Ehrhart theory was introduced by the author in [52] both in order to extend and reprove many results in Ehrhart theory, the study of enumeration of lattice points in polytopes, and in order to give explicit computations of motivic integrals on toric stacks [53]. We present a generalization of this theory below using an extension of Stanley’s theory of subdivisions [51] presented in [29]. For brevity, we will only present the relevant results and refer the reader to [29] and [52] for details of proofs.

4.1 Eulerian posets

We first recall some basic notions on Eulerian posets. Consider a finite poset B containing a minimal element \(\widehat{0}\) and a maximal element \(\widehat{1}\). For any pair \(z \le x\) in B, we can consider the interval \([z,x] = \{ y \in B \mid z \le y \le x \}\). Assume that B is graded in the sense that for every \(x \in B\), every maximal chain in the interval \([\widehat{0},x]\) has the same length \(\rho (x)\). We call \(\rho : B \rightarrow \mathbb {N}\) the rank function of B, and call \(\rho (\widehat{1})\) the rank of B. Then B is Eulerian if every interval [z, x] with \(z < x\) has as many elements of odd rank as even rank.

Example 4.1

The poset of faces of a polytope P (including the empty face) is an Eulerian poset under inclusion, with \(\rho (Q) = \dim Q + 1\), for any face Q of P. Throughout this paper, the empty polytope has dimension\(-1\).

Example 4.2

If B is a poset, then \(B^*\) is the poset with the same elements as B and all orderings reversed. In particular, B is Eulerian if and only if \(B^*\) is Eulerian.

The g-polynomial of an Eulerian poset is defined recursively and was introduced by Stanley [51, Corollary 6.7].

Definition 4.3

Let B be an Eulerian poset of rank n. If \(n = 0\), then \(g(B;t) = 1\). If \(n > 0\), then g(B; t) is the unique polynomial of degree strictly less the n / 2 satisfying

4.2 Polyhedral subdivisions

A polyhedral subdivision of a polytope \(P\subseteq \mathbb {R}^n\) is a subdivision of P into a finite number of polytopes such that the intersection of any two polytopes is a (possibly empty) face of both. A polyhedral subdivision is a rational (respectively, lattice) subdivision if each polytope has rational (respectively, integral) vertices. A natural class of lattice polyhedral subdivisions are the regular lattice subdivisions. They are induced by a function \(\omega : S \rightarrow \mathbb {Z}\), called an integral height function, for some finite subset \(S \subseteq P \cap \mathbb {Z}^n\) containing the vertices of P. The cells of the subdivision are the projections of the bounded faces of the convex hull \({\text {UH}}\) of \(\{ (u, \lambda ) \mid u \in S, \lambda \ge \omega (u) \} \subseteq \mathbb {R}^n \times \mathbb {R}\). The bounded faces of \({\text {UH}}\) form the graph of a function \(\nu : P \rightarrow \mathbb {R}\), called the convex graph of \(\omega \). Note that \(\nu |_S\) is not necessarily the same as \(\omega |_S\). Let \(\mathcal {S}(\nu )\) denote the regular lattice polyhedral subdivision induced by \(\omega \). Then \(\nu \) is convex and piecewise \(\mathbb {Q}\)-affine with respect to \(\mathcal {S}(\nu )\), and takes integer values on the vertices of \(\mathcal {S}(\nu )\). Conversely, if \(\nu : P \rightarrow \mathbb {R}\) is a convex function that is piecewise \(\mathbb {Q}\)-affine with respect to a lattice polyhedral subdivision \(\mathcal {S}\), such that \(\nu \) takes integer values on the vertices of \(\mathcal {S}\), then the restriction of \(\nu \) to the vertices of \(\mathcal {S}\) is an integral height function such that \(\nu \) is the corresponding convex graph and \(\mathcal {S}\) is a refinement of \(\mathcal {S}(\nu )\). For more details, including the general definition of a regular polyhedral subdivision, see [31, 36].

By abuse of notation, we write \(\mathcal {S}\) for the poset of faces (including the empty face) of a polyhedral subdivision \(\mathcal {S}\) of P. Let \([\emptyset , P]\) denote the face poset of P. In what follows, we will consider the function

where \(\sigma (\emptyset ) = \emptyset \), and for a nonempty cell \(F \in \mathcal {S}\), \(\sigma (F)\) is the smallest face of P containing F. As a poset, the link\({{\mathrm{lk}}}_\mathcal {S}(F)\) of F in \(\mathcal {S}\) consists of all cells \(F'\) of \(\mathcal {S}\) that contain F.

We will need the following two closely related polynomials. In the case \(F = \emptyset \), the local h-polynomial defined below was introduced by Stanley in [51, Corollary 7.7]. In the case of a triangulation of a simplex, it was introduced by Athanasiadis [1, 2], and the general case of a polyhedral subdivision of a polytope was first considered by Nill and Schepers [41], where it was called the ‘relative local h-polynomial’. A more general definition of the local h-polynomial appears in [29, Definition 4.1].

Definition 4.7

Let \(\mathcal {S}\) be a polyhedral subdivision of a polytope P. If F is a (possibly empty) cell of \(\mathcal {S}\), then the h-polynomial of \({{\mathrm{lk}}}_\mathcal {S}(F)\) is defined by

Example 4.8

If \(\mathcal {S}\) is the trivial subdivision of P, with cells of \(\mathcal {S}\) given by the faces of P, then one can verify that \(h({{\mathrm{lk}}}_{\mathcal {S}}(Q);t) = g([Q,P];t)\) and \(l_P(\mathcal {S},Q;t) = 0\) whenever Q is a (possibly empty) proper face of P.

We state some properties of these polynomials below. The nonnegativity of the g-polynomial and h-polynomial and the local h-polynomial when \(F = \emptyset \) is due to Stanley [51, Theorem 5.2,Theorem 7.9].

Remark 4.9

[29, Theorem 6.1] Let \(\mathcal {S}\) be a rational polyhedral subdivision of a lattice polytope P. For any cell \(F \in \mathcal {S}\) and face Q of P, the polynomials \(g([Q,P]^*;t)\), g([Q, P]; t), \(h({{\mathrm{lk}}}_\mathcal {S}(F);t)\) and \(l_P(\mathcal {S},F;t)\) have nonnegative coefficients. We have the symmetry

4.3 Weighted Ehrhart theory

Fix a lattice polytope P with respect to a lattice M. After possibly replacing M with a sublattice, we may assume that \(\dim P = \dim M_\mathbb {R}\). Under an isomorphism \(M \cong \mathbb {Z}^n\), the normalized volume\({{\mathrm{Vol}}}(P) \in \mathbb {Z}_{> 0}\) of P with respect to M is n! times the Euclidean volume of P. Let \(\nu : P \rightarrow \mathbb {R}\) be the convex graph of an integral height function on P, with corresponding regular lattice polyhedral subdivision \(\mathcal {S}(\nu )\) of P.

Remark 4.11

The setup above is slightly different than the ‘weight 0’ setup in [52], although the same arguments hold. We explain the relationship below. In [52], we consider a stacky fan \((N,\Sigma ,\{ b_i \})\), where \(\Sigma \) is a full-dimensional simplicial fan with respect to a lattice N, and \(\{ b_i \}\) is a choice of a nonzero lattice point on each ray of \(\Sigma \). We let \(\psi \) be the piecewise \(\mathbb {Q}\)-linear function with respect to \(\Sigma \) satisfying \(\psi (b_i) = 1\), and let \(Q = \{ v \in N_\mathbb {R}\mid \psi (v) \le 1 \}\). If Q is a convex polytope, then we may set \(M = N\), \(P = Q\) and \(\nu = \psi \) above to recover this setup. Here \(\Sigma \) induces a triangulation \(\mathcal {S}\) of P refining \(\mathcal {S}(\nu )\) such that \(\nu \) takes integer values on the vertices of \(\mathcal {S}\). We also note that the definition of the weighted \(h^*\)-polynomial below is slightly different (although equivalent) to that in [52] (see Example 5.13).

Let C be the cone over \(P \times \{ 1\}\) in \(M_\mathbb {R}\times \mathbb {R}\), and extend \(\nu \) by linearity to a piecewise \(\mathbb {Q}\)-linear function on C. For a nonnegative integer m, we will often identify mP with \(C \cap (M_\mathbb {R}\times \{ m \})\) and \(M_\mathbb {R}\) with \(M_\mathbb {R}\times \{ m \}\). We define a weight function

for every nonnegative integer m. As in Remark 2.4, the corresponding polynomial with coefficients in \(\mathbb {Z}\) is the usual Ehrhart polynomial of P [20, 21], defined by \(f(P;m) = \#(mP \cap M )\) for every nonnegative integer m. By [52, Theorem 3.7], \(f(P,\nu ;m)\) is a polynomial in m of degree \(\dim P\), and for every positive integer m,

The latter result is known as weighted Ehrhart reciprocity, since it reduces to the classical Ehrhart reciprocity: \(f(P;-m) = \#({{\mathrm{Int}}}(mP) \cap M )\) for every positive integer m [22]. In fact, it is shown in [52, Remark 3.10] that weighted Ehrhart reciprocity generalizes Ehrhart reciprocity for rational polytopes. Let \(\mathcal {S}\) be a lattice polyhedral decomposition refining \(\mathcal {S}(\nu )\) such that \(\nu \) takes integer values on the vertices of \(\mathcal {S}\). For each maximal cell \(F \in \mathcal {S}\), let \(m_F\) be the minimal positive integer such that \(m_F \nu |_{F}\) is an affine function with respect to M. Then one verifies that the leading coefficient of \(f(P,\nu ;m)\) as a polynomial in m is given by:

where \({{\mathrm{Vol}}}(F)\) is the normalized volume of F and \({{\mathrm{Vol}}}(F)/m_F\) is a positive integer. Define the weighted\(h^*\)-polynomial\( h^*(P,\nu ;u) \in \mathbb {Z}[\mathbb {Q}/\mathbb {Z}][u]\) by

When P is the empty polytope, we set \(h^*(P,\nu ;u) = 1\). As in Remark 2.4, the corresponding polynomial with coefficients in \(\mathbb {Z}\) is the usual \(h^*\)-polynomial\(h^*(P;u) \in \mathbb {Z}[u]\) of P introduced by Stanley in [50]. We define the local weighted\(h^*\)-polynomial\(l^*(P,\nu ;u) \in \mathbb {Z}[\mathbb {Q}/\mathbb {Z}][u]\) by

Here the sum runs over all faces of P including the empty face. When P is the empty polytope, we set \(l^*(P,\nu ;u) = 1\). As in Remark 2.4, the corresponding polynomial with coefficients in \(\mathbb {Z}\) is the usual local\(h^*\)-polynomial\(l^*(P;u) \in \mathbb {Z}[u]\) of P, introduced by Stanley in [51, Example 7.13] and later independently by Borisov and Mavlyutov in [5]. By Theorem 4.6 and Remark 4.9, the weighted \(h^*\)-polynomial can be recovered as a ‘nonnegative’ combination of local weighted \(h^*\)-polynomials:

Remark 4.14

One may verify that weighted Ehrhart reciprocity is equivalent to the following statements about the weighted \(h^*\)-polynomial and local weighted \(h^*\)-polynomial, respectively (cf. [29, Lemma 7.11]). Firstly, the weighted \(h^*\)-polynomial is acceptable in the sense of Stanley [51]. That is,

Remark 4.16

Let \(\mathcal {S}\) be a lattice polyhedral subdivision of P that refines \(\mathcal {S}(\nu )\) such that \(\nu \) takes integer values on the vertices of \(\mathcal {S}\). Then it follows from the definitions that for every positive integer m,

Remark 4.17

One may always choose a lattice triangulation \(\mathcal {S}\) of P that refines \(\mathcal {S}(\nu )\) such that \(\nu \) takes integer values on the vertices of \(\mathcal {S}\). In fact, there exists a choice with the same vertices as \(\mathcal {S}(\nu )\) (see, for example, [42]). Then Remark 4.16 together with Example 4.13 gives explicit formulas that one may use to compute \(h^*(P,\nu ;u)\) and \(l^*(P,\nu ;u)\) in practice. In particular, using Remark 4.9, it follows that all integer coefficients in \(h^*(P,\nu ;u)\) and \(l^*(P,\nu ;u)\) are nonnegative.

for \(1 \le i \le \dim P\). This follows by the proof of [29, Theorem 7.20]. Explicitly, consider a regular, lattice polyhedral subdivision \(\mathcal {S}\) of P that refines \(\mathcal {S}(\nu )\) such that the lattice points of weight 0 in P are precisely the vertices of \(\mathcal {S}\). Then \(\nu \) takes integer values on the vertices of \(\mathcal {S}\), and every positive-dimensional cell in \(\mathcal {S}\) contains no interior lattice points of weight 0. By Remark 4.16 and Example 4.15,

4.4 Mixed invariants

We introduce our main combinatorial invariants below. As in Remark 2.4, the corresponding invariants with coefficients in \(\mathbb {Z}\) first appeared in [29]. A geometric description of these invariants is provided in Corollary 5.12.

We summarize the main properties of these invariants in the following theorem. They may all be deduced from the definitions and the properties in the previous sections, and hence we omit the proof. We refer the reader to [29, Theorem 9.2, Theorem 9.3,Theorem 9.4] for a similar statement for the corresponding invariants with coefficients in \(\mathbb {Z}\).

Theorem 4.21

Let P be a lattice polytope and let \(\nu : P \rightarrow \mathbb {R}\) be the convex graph of an integral height function on P, with corresponding regular lattice polyhedral subdivision \(\mathcal {S}(\nu )\) of P. Then the following holds:

(1)

The weighted refined limit mixed \(h^*\)-polynomial satisfies the following symmetries:

The local weighted limit mixed \(h^*\)-polynomial specializes to the local weighted \(h^*\)-polynomial:

$$\begin{aligned} l^*(P,\nu ;u,1) = l^*(P,\nu ;u). \end{aligned}$$

(5)

The degree of \(h^*(P,\nu ;u,v,w)\) as a polynomial in w is at most \(\dim P + 1\). Moreover, the coefficient of \(w^{\dim P + 1}\) is the local weighted limit mixed \(h^*\)-polynomial \(l^*(P, \nu ; u,v)\). We have symmetries:

Remark 4.22

As in [29, Section 9], after fixing r, one may view the coefficients \(h^*_{p,q,r}(P,\nu )\) in a diamond, for example, by placing \(h^*_{p,q,r}(P,\nu )\) at point \((q - p, p + q)\) in \(\mathbb {Z}^2\). The resulting diamond of coefficients is symmetric with respect to its symmetry about the horizontal axis and symmetric up to conjugation with respect to its symmetry about the vertical axis. Fixing \([k] \in \mathbb {Q}/\mathbb {Z}\), the coefficients of [k] in each vertical strip of the diamond form a symmetric, unimodal sequence. The coefficients of [0] in each horizontal strip of the diamond satisfy the following lower bound theorem: the first entry is a lower bound for the other entries.

When \(\dim P = 3\), this gives an explicit description of all terms in \(h^*(P,\nu ;u,v,w)\) except \(h^*_{1,1,2}\). The remaining term can be deduced from the expression for \(h^*(P,\nu ;1,1,1) = h^*(P,\nu ;1)\) in Example 4.12.

5 Degenerations of hypersurfaces

In this section, we use Theorem 3.2 together with the combinatorics of Sect. 4 to prove a formula for the equivariant refined limit Hodge–Deligne polynomial of a schön hypersurface of a torus. We also give an equivalent formula for the equivariant refined limit Hodge–Deligne polynomial associated with the intersection cohomology of a canonical compactification of such a hypersurface. The corresponding results for invariants with coefficients in \(\mathbb {Z}\) appear in [30]. We will continue with the notation of Sects. 2 and 4.

Tropical geometry of hypersurfaces reduces to the study of Newton polytopes and polyhedral subdivisions [31, 46]. In particular, \({{\mathrm{Trop}}}(X^\circ )\) admits a polyhedral structure \(\Sigma \) with cells \(\gamma \) in inclusion-reversing correspondence with positive-dimensional cells F of \(\mathcal {S}(\nu _f)\), such that the bounded cells of \(\Sigma \) correspond to the cells of \(\mathcal {S}(\nu _f)\) not contained in the boundary of P. For example, if F is a maximal dimensional cell in \(\mathcal {S}(\nu _f)\), then \(\nu _f|_F\) is a \(\mathbb {Q}\)-affine function and the corresponding \(\mathbb {Q}\)-linear function is an rational point in \(N_\mathbb {Q}\), also denoted \(\nu _f|_F\). Then \(\gamma = -\nu _f|_F\) is a vertex in \(\Sigma \).

Let \(M_F\) denote the intersection of M with the translation of the affine span of F to the origin. Then equation in the right-hand side of (7) determines a schön complex hypersurface \(V_F^\circ \in {{\mathrm{Spec}}}\mathbb {C}[M_F]\), and \(\nu _f|_F\) determines a \(\mathbb {Q}\)-linear function on \((M_F)_\mathbb {Q}\). Multiplication by \(\exp (-2\pi \sqrt{-1} \nu _f|_F)\) determines a good \(\widehat{\mu }\)-action on \(V_F^\circ \). In this case, Theorem 3.2 translates into the following corollary.

Corollary 5.3

Let \(X^\circ \subseteq (\mathbb {K}^*)^n\) be a schön hypersurface, with associated Newton polytope and convex graph of an integral height function \((P,\nu )\) and \(\dim P = n\). Then the motivic nearby fiber of \(X^\circ \) is given by

Here \(\sigma (F)\) denotes the smallest face of P containing F, and \(V_{F}^\circ \) is a schön complex hypersurface of a torus with good \(\widehat{\mu }\)-action induced by multiplication by \(\exp (-2\pi \sqrt{-1} \nu |_F)\).

Remark 5.4

In fact, the sum in Corollary 5.3 runs only over the positive-dimensional faces in \(\mathcal {S}(\nu )\), since when F is a vertex of \(\mathcal {S}(\nu )\), \(V_{F}^\circ = \emptyset \). Here we follow the convention that \([V] = 0 \in K_0^{\widehat{\mu } }({{\mathrm{Var}}}_\mathbb {C})\) if V is empty.

We will also prove an equivalent statement involving intersection cohomology. We will use middle perversity throughout. As explained in detail in [30, Section 6], given a projective variety X over \(\mathbb {K}\), for each of the geometric invariants described in Section 2.3, one may consider the corresponding invariant with respect to intersection cohomology rather than cohomology, e.g., the equivariant refined limit Hodge–Deligne polynomial \(E_{{{\mathrm{int}}}}(X_\infty , \widehat{\mu };u,v,w)\) associated with the intersection cohomology of X. The key benefit of using intersection cohomology is the fact that the intersection cohomology groups of a complex projective variety admit a pure (rather than just mixed) Hodge structure.

Lemma 5.10

It is well known that all interesting cohomology above occurs in middle dimension. We review the relevant facts below and refer the reader to [30, Sections 5.3,6] for details. We have weak Lefschetz theorems stating that the Gysin maps \(H^{m}_c(X^\circ _{{{\mathrm{gen}}}})\rightarrow H^{m+2}_c((\mathbb {C}^*)^n)\) and \(IH^{m}(X_{{{\mathrm{gen}}}})\rightarrow IH^{m+2}(\mathbb {P}(\Delta _P)_\mathbb {C})\) are surjective for \(m\ge n - 1\) and an isomorphism if \(m > n - 1\). One may deduce from Theorem 5.8 that the monodromy maps associated with the cohomology with compact supports and intersection cohomology of \((\mathbb {K}^*)^n\) and \(\mathbb {P}(\Delta _P)_\mathbb {K}\), respectively, are trivial. Since \(X^\circ _{{{\mathrm{gen}}}}\) is affine, \(H^{m}_c(X^\circ _{{{\mathrm{gen}}}}) = 0\) for \(m < n - 1\). Also, Poincaré duality for intersection cohomology means that the Gysin isomorphism above determines \(IH^{m}(X_{{{\mathrm{gen}}}})\) for \(m < n - 1\). We let \(H^{n - 1}_{{{\mathrm{prim}}},c}(X^\circ _{\infty })\) and \(IH^{n - 1}_{{{\mathrm{prim}}}}(X_{\infty })\) denote the kernels of the above Gysin maps with induced filtrations. These groups constitute the ‘interesting’ cohomology. Consider the contributions of these groups (up to a sign) to the equivariant refined limit Hodge–Deligne polynomial:

Before proceeding to the proof of Theorem 5.7 and Theorem 5.9, we will need the following theorem. In fact, it is a consequence of the proof that the theorem below is a special case of Theorem 5.7 and Theorem 5.9. An algorithm to compute the equivariant Hodge–Deligne polynomial below is given in [39, Section 2], extending an algorithm of Danilov and Khovanskiĭ in the nonequivariant setting [14]. The theorem below generalizes a formula of Borisov and Mavlyutov in the nonequivariant setting [5].

If \(X = \cup _{Q \subseteq P} X^\circ _Q\) denotes the stratification in the statement of Theorem 5.8, then \(X^\circ _Q\) is defined by an equation of the form \(\{ \sum _{u \in Q \cap M} \lambda _u t^{\nu (u)} x^u = 0 \} \). It follows that if we restrict Theorem 5.7 to all \(X^\circ \) that appear from complex schön hypersurfaces \(V^\circ \) as above, then Theorem 5.7, Theorem 5.9 and (9) and (10) are all still equivalent when restricted to this class of varieties over \(\mathbb {K}\).

Here the last equality follows from (7) in Theorem 4.21. This establishes (10) as desired.

\(\square \)

Using Remark 5.1 and the above proof, we immediately deduce the following geometric description of the combinatorial invariants introduced in Sect. 4. As in Remark 2.4, the corresponding result for invariants with coefficients in \(\mathbb {Z}\) can be found in [30, Corollary 5.11,Corollary 6.3].

Corollary 5.12

Let P be a full-dimensional lattice polytope in a lattice M and let \(\nu : P \rightarrow \mathbb {R}\) be the convex graph of an integral height function on P. Let \(X^\circ = \{ f = 0 \} \subseteq {{\mathrm{Spec}}}\mathbb {K}[M]\) be a schön hypersurface with Newton polytope P such that \(\nu _f = \nu \). Let X denote the closure of \(X^\circ \) in the projective toric variety over \(\mathbb {K}\) corresponding to the normal fan of P. Then

Example 5.13

Let P be a lattice polytope in a lattice N such that P contains the origin in its relative interior, and let \(\{ b_i \}\) denote the vertices of P. Let \(\nu : P \rightarrow \mathbb {R}\) be the piecewise \(\mathbb {Q}\)-affine function with value 0 at the origin and value 1 on the boundary of P. For an explicit example, see Example 4.19. Let \(X^\circ = \{ f = 0 \} \subseteq {{\mathrm{Spec}}}\mathbb {K}[N]\) be a schön hypersurface with Newton polytope P such that \(\nu _f = \nu \). For example, if g is a schön complex polynomial with Newton polytope P, then take \(f = tg - \lambda \) for a generic choice of \(\lambda \in \mathbb {C}^*\). Define the primitive spectrum of f to be:

This was the definition of the weighted \(h^*\)-polynomial given in [52]. Note that the \(h^*\)-polynomial \(h^*(P;t)\) is obtained from \(\widetilde{h}(P; t)\) by rounding up the exponents of t to the nearest integer. By [52, Corollary 2.12], we have the symmetry:

Consider the complete fan \(\Sigma '\) in \(N_\mathbb {R}\) with cones given by the cones over the proper faces of P, and let \(\Sigma \) be a simplicial fan refinement of \(\Sigma '\) with the same rays. Then the vertices \(\{ b_i \}\) of P constitute a choice of a nonzero lattice point on each ray of \(\Sigma \). As in Remark 4.11, the triple \(\varvec{\Sigma }= (N,\Sigma ,\{ b_i \})\) is a stacky fan and we may consider the corresponding Deligne-Mumford stack \({\mathcal {X}} = {\mathcal {X}}(\varvec{\Sigma })\) with coarse moduli space the toric variety associated with the fan \(\Sigma \) [4]. The theory of orbifold cohomology, developed by Chen and Ruan [11, 12], associates to \({\mathcal {X}}\) a finite-dimensional \({\mathbb {Q}}\)-algebra \(H_{{{\mathrm{orb}}}}^{*}({\mathcal {X}}, {\mathbb {Q}})\), graded by \({\mathbb {Q}}\).

By [52, Theorem 4.3], together with Corollary 5.12, we see that the primitive spectrum of f coincides with the Betti polynomial of \({\mathcal {X}}\):

6 Applications to the monodromy of complex polynomials

In this section, we apply Theorem 5.7 to deduce explicit combinatorial equations for some important invariants associated with the monodromy of complex polynomials. Recall that we identify the group \(\mathbb {Q}/\mathbb {Z}\) with the group \({\mathbb {S}}^1_\mathbb {Q}\) of rational points on the circle \(\{ z \in \mathbb {C}\mid |z| = 1 \}\), sending \([k] \in \mathbb {Q}/\mathbb {Z}\) to \(\alpha = e^{2 \pi \sqrt{-1}k} \in {\mathbb {S}}^1_\mathbb {Q}\). We will continue with the notation of Sects. 2, 4 and 5, and set \(M = \mathbb {Z}^n\).

6.1 Degenerations of hypersurfaces in affine space

We first consider applications of the results of Sect. 5 for families of hypersurfaces of affine space. Let \(X^\circ \subseteq (\mathbb {K}^*)^n\) be a schön hypersurface, with associated Newton polytope and convex graph of an integral height function \((P,\nu )\) and \(\dim P = n\). Assume that \(P \subseteq \mathbb {R}_{\ge 0}^n\), and for each (possibly empty) subset S of \(\{1,\ldots ,n\}\), let \(\mathbb {R}^S = \{ (v_1,\dots ,v_n) \mid v_i = 0 \text { if } i \notin S \}\), \(\mathbb {Z}^S = \mathbb {R}^S \cap \mathbb {Z}^n\) and \(P^S = P \cap \mathbb {R}^S\). We assume that P is convenient in the sense that \(\dim P^S\) equals the cardinality |S| of S for every subset S of \(\{1,\ldots ,n\}\). Equivalently, P contains the origin and has nonzero intersection with each ray through a coordinate vector. We call a face (including the empty face) of P that does not contain the origin, a face at infinity, and write \(P_\infty \) for the union of faces at infinity of P. Then the faces of P are precisely the faces at infinity together with \(\{ P^S \mid S \subseteq \{ 1, \ldots , n\} \}\). Let X denote the closure of \(X^\circ \) in \(\mathbb {K}^n\). Then \(X_{{{\mathrm{gen}}}} \subseteq \mathbb {C}^n\) is a smooth hypersurface, and as in Sect. 5, one has a weak Lefschetz result (see, for example, [14, Corollary 3.8]) stating that \(H^{2(n - 1)}_c(X_{{{\mathrm{gen}}}}) = \mathbb {C}\) with trivial monodromy action and the only other nonzero cohomology is in middle dimension \(H^{n - 1}_c(X_{{{\mathrm{gen}}}})\). By Corollary 5.3, the motivic nearby fiber of X is given by

Specializing by setting \(w = 1\) gives a formula for the equivariant limit Hodge–Deligne polynomial. We may specialize further by setting \(v = 1\) and then \(u = 1\). In particular, by Example 4.12, we have the following formula for the eigenvalues (with multiplicity) of the action of monodromy on the cohomology of \(X_{{{\mathrm{gen}}}}\):

where \(m_F\) is the minimal positive integer such that \(m_F \nu |_{F}\) is an affine function with respect to \(\mathbb {Z}^S\). Here \({{\mathrm{Vol}}}(F)\) is the normalized volume of F, and \({{\mathrm{Vol}}}(F)/m_F\) is a positive integer.

Remark 6.1

With the setup above, do not assume that P is convenient, and instead only assume that P contains the origin. Then we may not apply the same weak Lefschetz argument. However, we still obtain the following formulas for the motivic nearby fiber and equivariant refined limit Hodge–Deligne polynomial:

Example 6.2

With the notation above, let Q be a face at infinity and consider G(Q; t). By definition, G(Q; t) has degree strictly less than \((\dim P - \dim Q)/2\). If \({{\mathrm{Int}}}(Q) \subseteq \mathbb {R}_{> 0}^n\), then \(G(Q;t) = g([Q,P];t)\) has constant term 1. Otherwise, G(Q; t) has no constant term. Using Example 4.4, when \(Q = \emptyset \), we compute that the linear coefficient of G(Q; t) is the number of vertices of P contained in \(\mathbb {R}_{> 0}^n\).

where every term in \(\beta (u,v)\) has combined degree in u and v at least 4.

Observe that every term in F(uw, vw) has combined degree in u and v equal to its degree in w. Hence, the only contribution above corresponding to a nontrivial action of monodromy is \((-w)^{n+ 1} l^*(P,\nu ;u,v)\). We conclude that we have the following corollary.

Corollary 6.3

Let \(X^\circ \subseteq (\mathbb {K}^*)^n\) be a schön hypersurface, with associated Newton polytope and convex graph of an integral height function \((P,\nu )\) and \(\dim P = n\). Let \(X \subseteq \mathbb {K}^n\) denote the closure of \(X^\circ \) in \(\mathbb {K}^n\). Assume that \(P \subseteq \mathbb {R}_{\ge 0}^n\) is convenient, and the restriction of \(\nu \) to every face at infinity is constant. Then the action of monodromy is trivial on the graded pieces \(Gr^W_{r} H^{n - 1}_c (X_{{{\mathrm{gen}}}})\) of the Deligne weight filtration for \(r \ne n - 1\), and we have the following explicit combinatorial formula for the equivariant limit mixed Hodge numbers of \(Gr^W_{ n - 1} H^{n - 1}_c (X_{{{\mathrm{gen}}}})\):

In particular, the action of monodromy on \(Gr^W_{0} H^{1}_c (X_{{{\mathrm{gen}}}})\) is trivial and \(\dim Gr^W_{0} H^{1}_c (X_{{{\mathrm{gen}}}}) = b\). The equivariant limit mixed Hodge numbers of \(Gr^W_{1} H^{1}_c (X_{{{\mathrm{gen}}}})\) are given by

We present the following two examples of the above situation. Below we let \(f \in \mathbb {C}[x_1, \ldots , x_n]\) be a complex polynomial with Newton polytope \({{\mathrm{NP}}}(f)\). For every face Q of \({{\mathrm{NP}}}(f)\), we let \(\Delta _Q\) denote the convex hull of Q and the origin. We set \(P = \Delta _{{{\mathrm{NP}}}(f)}\).

We assume that f is convenient in the sense that \({{\mathrm{NP}}}(f)\) has nonzero intersection with each ray through a coordinate vector [34]. Recall that for a complex hypersurface of a torus, schön is also called nondegenerate. Let \(\Gamma _+(f) = {{\mathrm{NP}}}(f) + \mathbb {R}_{\ge 0}^n\) denote the Newton polyhedron of f. Observe that every bounded face of \(\Gamma _+(f)\) is a face of NP(f). We let \(\Gamma _f\) denote the union of the bounded faces of \(\Gamma _+(f)\).

Remark 6.6

For any proper face Q of \({{\mathrm{NP}}}(f)\) not containing the origin, consider a \(\mathbb {Q}\)-affine function \(\nu \) on \(\Delta _Q\) with value \(m \pm 1\) at the origin, and value m on Q, for some integer m. Consider \(l^*(Q,\nu ;u) \in \mathbb {Z}[\mathbb {Q}/\mathbb {Z}][u]\) as a sum of integer-valued polynomials indexed by \(\alpha \in \mathbb {Q}/\mathbb {Z}\). Then the coefficient of \(\alpha = 1\) is zero. To see this, one may for example use Remark 4.16 to reduce to the case when Q is a simplex, and then apply Example 4.13.

By (13), we have a formula for the eigenvalues (with multiplicity) of the action of monodromy on the cohomology of \(X_{{{\mathrm{gen}}}}\). Explicitly, \(\sum _{\alpha \in \mathbb {Q}/\mathbb {Z}} \dim H_{c}^{n - 1}(X_\infty )_\alpha \alpha \) is equal to

where \({{\mathrm{Vol}}}(Q)\) is the normalized volume of Q and m(Q) is the lattice distance of Q from the origin, i.e., m(Q) is the minimal positive integer such that \(m(Q) \nu _0|_{\Delta _Q}\) is an affine function with respect to the lattice given by intersecting M with the affine span of \(\Delta _Q\). When Q is empty, \({{\mathrm{Vol}}}(Q) = m(Q) = 1\) and \(\dim Q = -1\).

Example 6.8

(Monodromy at infinity) Assume that \(f \in \mathbb {C}[x_1, \ldots , x_n]\) is convenient. Assume that f is schön at infinity, meaning that \(\{ f|_Q = 0 \} \subseteq (\mathbb {C}^*)^n\) defines a smooth hypersurface whenever Q is a face of P at infinity. Then for a general choice of \(\lambda \in \mathbb {C}^*\), \(X^\circ = \{ ft = \lambda \} \subseteq (\mathbb {K}^*)^n\) defines a schön, convenient hypersurface with Newton polytope P, and \(\nu _\infty := \nu \) is the piecewise \(\mathbb {Q}\)-affine function on P with value 0 at the origin and value identically 1 on the faces at infinity of P. The cells of \(\mathcal {S}(\nu _\infty )\) are given by the union of \(\{ Q \mid Q \subseteq P_\infty \}\) and \(\{ \Delta _Q \mid Q \subseteq P_\infty \}\). By (11), we have the following equation for the motivic nearby fiber:

The motivic nearby fiber at infinity of a convenient polynomial f was introduced independently and from different perspectives in [39] and [45], and an explicit formula is given in [39, Theorem 5.3], which agrees with our formula for the motivic nearby fiber above. In particular, the motivic nearby fiber at infinity is equal to the usual motivic nearby fiber of X.

After specializing at \(v = w = 1\), one obtains a formula for the spectrum at infinity of f [47] that is equivalent to [39, Theorem 5.11]. Recall from (13) that specialization at \(u = v = w = 1\) yields a formula for the eigenvalues (with multiplicity) of the action of monodromy:

where \({{\mathrm{Vol}}}(Q)\) is the normalized volume of Q and m(Q) is the lattice distance of Q from the origin, i.e., m(Q) is the minimal positive integer such that \(m(Q) \nu _\infty |_{\Delta _Q}\) is an affine function with respect to the lattice given by intersecting M with the affine span of \(\Delta _Q\). When Q is empty, \({{\mathrm{Vol}}}(Q) = m(Q) = 1\) and \(\dim Q = -1\). The above is equivalent to a formula of Libgober and Sperber [35] for the zeta function at infinity of f.

In this case, the action of monodromy on \(Gr^W_{1} H^{1}_c (X_{{{\mathrm{gen}}}})\) has 16 Jordan blocks of size 1, 14 with eigenvalue 1 and 1 with eigenvalue \(\exp (2\pi \sqrt{-1}/3)\) and \(\exp (4\pi \sqrt{-1}/3)\), respectively, as well as 2 Jordan blocks of size 2 with eigenvalue 1. For monodromy at infinity, we have

6.2 Jordan block structure of monodromy at infinity

We continue with the notation of Example 6.8 above, and let \(f \in \mathbb {C}[x_1, \ldots , x_n]\) be convenient and schön at infinity, and let \(X = \{ ft = \lambda \} \subseteq \mathbb {K}^n\) for a general choice of \(\lambda \in \mathbb {C}^*\). Recall that for every face Q of the Newton polytope \({{\mathrm{NP}}}(f)\), we let \(\Delta _Q\) denote the convex hull of Q and the origin. Recall that \(P = \Delta _{{{\mathrm{NP}}}(f)}\), and \(P_\infty \) denotes the union of the faces at infinity, i.e., the faces of P not containing the origin. Recall that \(\nu _\infty \) is the piecewise \(\mathbb {Q}\)-affine function on P with value 0 at the origin and value identically 1 on the faces at infinity of P, and the cells of \(\mathcal {S}(\nu _\infty )\) are given by the union of \(\{ Q \mid Q \subseteq P_\infty \}\) and \(\{ \Delta _Q \mid Q \subseteq P_\infty \}\). Recall that \(\Delta ^\infty \) is the simplex \(P \cap \{ x_1 + \cdots + x_n = \epsilon \}\) for fixed \(\epsilon > 0\) sufficiently small, and \(\mathcal {S}_\infty \) denotes the regular, rational polyhedral subdivision of \(\Delta ^\infty \) obtained by intersecting \(\mathcal {S}(\nu _\infty )\) with \(\Delta ^\infty \).

Following [39], one may use results of Broughton and Sabbah to read off the Jordan normal form of the action of monodromy on \(H^{n - 1}_c (X_{{{\mathrm{gen}}}})\) from the equivariant limit Hodge–Deligne polynomial. We outline the argument below. By Poincaré duality, we may work with usual cohomology \(H^{n - 1} (X_{{{\mathrm{gen}}}})\) rather than cohomology with compact supports. We write \(H^{n - 1} (X_{{{\mathrm{gen}}}}) = H^{n - 1} (X_{{{\mathrm{gen}}}})_{= 1} \oplus H^{n - 1} (X_{{{\mathrm{gen}}}})_{\ne 1}\), where \(H^{n - 1} (X_{{{\mathrm{gen}}}})_{= 1}\) denotes the 1-eigenspace of \(H^{n - 1} (X_{{{\mathrm{gen}}}})\), and \(H^{n - 1} (X_{{{\mathrm{gen}}}})_{\ne 1}\) denotes the sum of the \(\alpha \)-eigenspaces, for \(\alpha \ne 1\). The induced monodromy weight filtration \(M_\bullet \) on \(H^{n - 1} (X_{{{\mathrm{gen}}}})\) may be written as \(M_\bullet = (M_{=1})_\bullet \oplus (M_{\ne 1})_\bullet \), where \((M_{=1})_\bullet \) and \((M_{\ne 1})_\bullet \) are the induced filtrations on \(H^{n - 1} (X_{{{\mathrm{gen}}}})_{= 1}\) and \(H^{n - 1} (X_{{{\mathrm{gen}}}})_{\ne 1}\), respectively. Let \(N = \log T_u\) denote the nilpotent operator where \(T_u\) is the unipotent part of monodromy acting on \(H^{n - 1} (X_{{{\mathrm{gen}}}})\), and consider the induced action of N on \(H^{n - 1} (X_{{{\mathrm{gen}}}})_{= 1}\) and \(H^{n - 1} (X_{{{\mathrm{gen}}}})_{\ne 1}\). As explained in detail in [39], using results of Broughton [9], we have the following special case of a deep result of Sabbah. Recall the definition of the weight filtration of a nilpotent operator from Definition 2.6.

Corollary 6.14

Let \(f \in \mathbb {C}[x_1, \ldots , x_n]\) be convenient and schön at infinity, and let \(J^\infty _{k,\alpha }\) be the number of Jordan blocks of size k with eigenvalue \(\alpha \) for the action of monodromy at infinity on \(H^{n - 1} (f^{-1}(t))\) for t fixed and sufficiently large. Then

Example 6.15

Let \(\partial \mathbb {R}^n_{\ge 0}\) denote the intersection of \(\mathbb {R}^n_{\ge 0}\) with the union of the coordinate hyperplanes. Using either Example 4.10 and Example 4.15, or Example 4.24 and Example 6.2, one computes the number of Jordan blocks with the largest and second largest possible size:

Example 6.17

Following on with Example 6.10, if \(f = \sum _{i = 1}^n x_i^{m_i} \in \mathbb {C}[x_1,\ldots ,x_n]\), for some \(m_i \in \mathbb {Z}_{>0}\), then the action of monodromy is semi-simple, and all Jordan blocks have size 1. With the notation of Example 6.10,

Recall that \(\Gamma _+(f) = {{\mathrm{NP}}}(f) + \mathbb {R}_{\ge 0}^n\) denotes the Newton polyhedron of f, and \(\Gamma _f\) denotes the union of the bounded faces of \(\Gamma _+(f)\). Recall that for every bounded face Q of \(\Gamma _+(f)\), \(\Delta _Q\) denotes the convex hull of Q and the origin. We assume that f is convenient, i.e., \(\Gamma _+(f)\) has nonzero intersection with each ray through a coordinate vector. We assume that f is schön at 0, meaning that \(\{ f|_Q = 0 \} \subseteq (\mathbb {C}^*)^n\) defines a smooth hypersurface whenever Q is a bounded face of \(\Gamma _+(f)\). Note that this is a weaker condition than the condition that f is schön in Example 6.7. Let \(P_+\) be the union of \(\{ \Delta _Q \mid Q \subseteq \Gamma _f \}\), and let \(\nu _0\) be the piecewise \(\mathbb {Q}\)-affine function on \(P_+\) with value 1 at the origin and value 0 on \(\Gamma _f\). Then the lattice polyhedral decomposition \(\mathcal {S}(\nu _0)\) of \(P_+\) has cells \(\{ \Delta _Q \mid Q \subseteq \Gamma _f \}\) and \(\{ Q \subseteq \Gamma _f \}\). Although we will not need this, we note that even though \(P_+\) is not convex, one can apply all the combinatorial constructions and results of Sect. 4. With the notation of Example 6.7, the following formula for the motivic Milnor fiber is given in [40, Theorem 4.3] (cf. (14)):

where \({{\mathrm{Vol}}}(Q)\) is the normalized volume of Q and m(Q) is the lattice distance of Q from the origin, i.e., m(Q) is the minimal positive integer such that \(m(Q) \nu _0|_{\Delta _Q}\) is an affine function with respect to the lattice given by intersecting M with the affine span of \(\Delta _Q\). When Q is empty, \({{\mathrm{Vol}}}(Q) = m(Q) = 1\) and \(\dim Q = -1\). This is equivalent to a famous formula of Varchenko [61] for the zeta function of monodromy of \(F_0\). As in Remark 2.4, we obtain a formula for \(\dim H^{n - 1}(F_0)\) originally due to Kouchnirenko [34].

Theorem 6.20

Let \(f \in \mathbb {C}[x_1,\ldots , x_n]\) be a complex polynomial such that \(f^{-1}(0)\) admits an isolated singularity at the origin. Assume further that f is convenient and schön at 0. With the notation above, we have the following formulas for the equivariant mixed Hodge numbers of the cohomology of the associated Milnor fiber \(F_0\):

where the sum runs over all (possibly empty) bounded faces of the Newton polyhedron of f.

We note that an algorithm to compute the equivariant mixed Hodge numbers of the cohomology of the associated Milnor fiber \(F_0\), as well as formulas in special cases, was given by Matsui and Takeuchi in [40], extending work of Danilov [13] and Tanabé [58].

Remark 6.21

There is a striking symmetry between the formulas for monodromy at infinity and monodromy of the Milnor fiber in Corollary 6.13 and Theorem 6.20, respectively. The existence of such a symmetry was first observed by Matsui and Takeuchi in [40].

The following corollary is immediate from the description of the weight filtration on the cohomology of \(F_0\) and Definition 2.6 and should be compared to Corollary 6.14. For every bounded face Q of \(\Gamma _f\), it follows from Remark 4.9 that we may write

Corollary 6.22

Let \(f \in \mathbb {C}[x_1,\ldots , x_n]\) be a complex polynomial such that \(f^{-1}(0)\) admits an isolated singularity at the origin. Assume further that f is convenient and schön at 0. Let \(J^0_{k,\alpha }\) be the number of Jordan blocks of size k with eigenvalue \(\alpha \) for the action of monodromy on the cohomology of the Milnor fiber \(H^{n - 1} (F_0)\). Then

where the sum runs over all (possibly empty) bounded faces of the Newton polyhedron of f.

Example 6.23

We have the following analogue of Example 6.15. Let \(\partial \mathbb {R}^n_{\ge 0}\) denote the intersection of \(\mathbb {R}^n_{\ge 0}\) with the union of the coordinate hyperplanes. Then we have the following formulas for the number of Jordan blocks with the largest and second largest possible size:

This reproves the main results of [40]. Namely, the first two equations are equivalent to [40, Theorem 1.1], and the second two equations are equivalent to [40, Theorem 1.2]. Also, the third equation is originally due to van Doorn and Steenbrink [60].

Example 6.24

We have the following analogue of Example 6.16. Consider the case when \(n = 2\). Then \(J^0_{1,1} = \#(\Gamma _f \cap \mathbb {Z}^{2}_{> 0})\), and

Example 6.25

We have the following analogue of Example 6.17. Following on with Example 6.10, if \(f = \sum _{i = 1}^n x_i^{m_i} \in \mathbb {C}[x_1,\ldots ,x_n]\), for some \(m_i \in \mathbb {Z}_{>0}\), then the action of monodromy is semi-simple, and all Jordan blocks have size 1. With the notation of Example 6.10,

Acknowledgements

The author would like to thank Anthony Henderson for some useful comments.

Declarations

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